Question: A box contains 4 white balls and 4 black balls.  I draw them out of the box, one at a time.  What is the probability that all of my draws alternate colors?
Solution: Let $B$ denote drawing a black ball and $W$ denote drawing a white ball.

There are two possible successful orders: $BWBWBWBW$ or $WBWBWBWB.$

There are $\binom{8}{4} = 70$ ways to arrange four $B$'s and four $W$'s, so the probability that a random arrangement is successful is $\dfrac{2}{70} = \boxed{\dfrac{1}{35}}$.

OR

We could also compute this based on the probabilities at each step that we draw a ball of the opposite color.  If we do that, we get $\frac47 \times \frac36 \times \frac35 \times \frac24 \times \frac23 \times \frac12 = \frac{144}{5040} = \boxed{\dfrac{1}{35}}$.